## A Generalized Klein Gordon Equation with a Closed System Condition for the Dirac-Current Probability Tensor

**Authors:** E. P. J. de Haas

By taking spin away from particles and putting it in the metric, thus following Dirac's vision, I start my attempt to formulate an alternative math-phys language, biquaternion based and incorporating Clifford algebra. At the Pauli level of two by two matrix representation of biquaternion space, a dual base is applied, a space-time and a spin-norm base. The chosen space-time base comprises what Synge called the minquats and in the same spirit I call their spin-norm dual the pauliquats. Relativistic mechanics, electrodynamics and quantum mechanics are analyzed using this approach, with a generalized Poynting theorem as the most interesting result. Then moving onward to the Dirac level, the M{\"o}bius doubling of the minquat/pauliquat basis allows me to formulate a generalization of the Dirac current into a Dirac probability/field tensor with connected closed system condition. This closed system condition includes the Dirac current continuity equation as its time-like part. A generalized Klein Gordon equation that includes this Dirac current probability tensor is formulated and analyzed. The usual Dirac current based Lagrangians of relativistic quantum mechanics are generalized using this Dirac probability/field tensor. The Lorentz transformation properties the generalized equation and Lagrangian is analyzed.

**Comments:** 42 Pages. Improved Lorentz transformation of the Dirac spinors.

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### Submission history

[v1] 2018-10-21 15:35:40

[v2] 2018-10-23 06:00:17

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